Opinion forming in Erdos-Renyi random graph and expanders

Assume for a graph G = (V, E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called the majority model, on the Erdos-Renyi random graph G(n,p) and regular expanders. First we consider the behavior of the majority model on G(n,p )with an initial random configuration, where each node is blue independently with probability p(b) and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely log n/n. Furthermore, we say a graph G is lambda-expander if the second-largest absolute eigenvalue of its adjacency matrix is lambda. We prove that for a Delta-regular lambda-expander graph if lambda/Delta is sufficiently small, then the majority model by starting from (1/2 - delta) n blue nodes (for an arbitrarily small constant delta > O) results in fully red configuration in sublogarithmically many rounds. Roughly speaking, this means the majority model is an "efficient" and "fast" density classifier on regular expanders. As a by-product of our results, we show that regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node Delta-regular Ramanujan graph if the initial number of blue nodes is s <= beta n, the number of blue nodes in the next round is at most cs/Delta for some constants c, beta > O. (C) 2019 Elsevier B.V. All rights reserved.